Truncated tetrapentagonal tiling
Nowadays, Truncated tetrapentagonal tiling is a topic that has gained great relevance in modern society. For years, Truncated tetrapentagonal tiling has been the subject of debate and discussion in different areas, whether in public policies, in the academic world or in people's daily lives. However, despite the importance that Truncated tetrapentagonal tiling has acquired, there are still many aspects that are little known or that generate controversy. In this article, we will explore different aspects of Truncated tetrapentagonal tiling in depth, analyzing its impact on society, its evolution over the years and the possible implications it has for the future.
| Truncated tetrapentagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.8.10 |
| Schläfli symbol | tr{5,4} or |
| Wythoff symbol | 2 5 4 | |
| Coxeter diagram |    or  
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| Symmetry group | , (*542) |
| Dual | Order-4-5 kisrhombille tiling |
| Properties | Vertex-transitive |
In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1,2{4,5} or tr{4,5}.
Symmetry
There are four small index subgroup constructed from by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
A radical subgroup is constructed , index 10, as , (5*2) with gyration points removed, becoming orbifold (*22222), and its direct subgroup +, index 20, becomes orbifold (22222).
| Small index subgroups of | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Index | 1 | 2 | 10 | ||||||||
| Diagram | 
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| Coxeter (orbifold) |
= (*542) |
=  =   (*552) |
=  (5*2) |
=    (*22222) | |||||||
| Direct subgroups | |||||||||||
| Index | 2 | 4 | 20 | ||||||||
| Diagram | 
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| Coxeter (orbifold) |
+ = (542) |
+ = =   (552) |
+ = (22222) | ||||||||
Related polyhedra and tiling
| *n42 symmetry mutation of omnitruncated tilings: 4.8.2n | ||||||||
|---|---|---|---|---|---|---|---|---|
| Symmetry *n42 |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||
| *242 |
*342 |
*442 |
*542 |
*642 |
*742 |
*842 ... |
*∞42 | |
| Omnitruncated figure |
 4.8.4 |
 4.8.6 |
 4.8.8 |
4.8.10 |
 4.8.12 |
 4.8.14 |
 4.8.16 |
 4.8.∞ |
| Omnitruncated duals |
 V4.8.4 |
 V4.8.6 |
 V4.8.8 |
 V4.8.10 |
 V4.8.12 |
 V4.8.14 |
 V4.8.16 |
 V4.8.∞ |
| *nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry *nn2 |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||||||||
| *222 |
*332 |
*442 |
*552 |
*662 |
*772 |
*882 ... |
*∞∞2 | |||||||
| Figure | 
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| Config. | 4.4.4 | 4.6.6 | 4.8.8 | 4.10.10 | 4.12.12 | 4.14.14 | 4.16.16 | 4.∞.∞ | ||||||
| Dual | 
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| Config. | V4.4.4 | V4.6.6 | V4.8.8 | V4.10.10 | V4.12.12 | V4.14.14 | V4.16.16 | V4.∞.∞ | ||||||
| Uniform pentagonal/square tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: , (*542) | +, (542) | , (5*2) | , (*552) | ||||||||
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| {5,4} | t{5,4} | r{5,4} | 2t{5,4}=t{4,5} | 2r{5,4}={4,5} | rr{5,4} | tr{5,4} | sr{5,4} | s{5,4} | h{4,5} | ||
| Uniform duals | |||||||||||
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| V54 | V4.10.10 | V4.5.4.5 | V5.8.8 | V45 | V4.4.5.4 | V4.8.10 | V3.3.4.3.5 | V3.3.5.3.5 | V55 | ||
See also
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
- Coxeter, H. S. M. (1999). "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. ISBN 0-486-40919-8. LCCN 99035678.
External links
- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
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